Problem #6: Explore problems of maximization such as the lidless box formed from a 5x8 sheet with a square removed from each corner.

Before exploring this problem with a spreadsheet, let us consider what this lidless box may look like:

Knowing that the shortest side of the sheet is 5 units, we can determine that the value for x can be no greater than 2.5 units. We can also infer that is we are making a lidless box, that the side length of the square, x, cut out from each corner can also be considered the height of the box. Then, for the purpose of our spreadsheet, we will label x as the height, 5-2x as the width, and 8-2x as the length.

It seems that 18 cubic units is the maximum volume for the box, implying that the side length for the square cut outs is 1 unit. We can continue exploring this idea graphically with Excel spreadsheets, because they say a picture is worth a thousand words.

First, we can plot the data on a scatterplot. The values for the height along the x-axis and the values for the volume along the y-axis. Once again, we notice 18 cubic units seems to be the maximum volume.

Or, we could plot the data using a bar graph. Once again, the height is along the horizontal axis and the volume is along the vertical axis with the maximum volume at 18 cubic units.

The exploration can continue by including an algebraic function and evaluating the corresponding graph. By representing the height of the box as x, the width of the box as 5-2x, and the length of the box as 8-2x, we can multiply these expressions together to find an algebraic expression for the volume of the box that is variable due to the height. This expression can be graphed using graphing calculator.

We can see that the graph has a relative maximum at the point (1, 18). Starting with spreadsheets is an excellent way to introduce maxization problems. Our investigations have shown us that the maximum value of the box made from a 5x8 sheet with squares cut off at the corners is 18 cubic units. The 18 cubic units are formed by the lidless box having a height of 1 unit, a width of 3 units, and a length of 6 units. Therefore, when making this box you must cut squares for the corners of the 5x8 sheet that have a side length of 1 unit.